Theorem toponcom 23279

Description: If 𝐾 is a topology on the base set of topology 𝐽, then 𝐽 is a topology on the base of 𝐾. (Contributed by Mario Carneiro, 22-Aug-2015.)

Assertion

Ref	Expression
toponcom	⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

Proof of Theorem toponcom

Step	Hyp	Ref	Expression
1		toponuni 23265	. . . 4 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐽 = ∪ 𝐾)
2	1	eqcomd 2769	. . 3 ⊢ (𝐾 ∈ (TopOn‘∪ 𝐽) → ∪ 𝐾 = ∪ 𝐽)
3	2	anim2i 629	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
4		istopon 23263	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐾) ↔ (𝐽 ∈ Top ∧ ∪ 𝐾 = ∪ 𝐽))
5	3, 4	sylibr 237	1 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ (TopOn‘∪ 𝐽)) → 𝐽 ∈ (TopOn‘∪ 𝐾))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∪ cuni 4870  ‘cfv 6561  Topctop 23244  TopOnctopon 23261

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-topon 23262

This theorem is used by:  toponcomb  23280  kgencn3  23910